1. PM2.5 map and Asthma map

PM2.5 map

Of 1580 tracts in Bay Area, about 10% tracts’ PM2.5 are more than 9.08 µg/m3 over three years (2015 to 2017)

Asthma map

Of 1580 tracts in Bay Area, about 10% tracts’ rate of ED visits for asthma are more than 104 per 10,000 (averaged over 2015-2017)

There are 42 tracts with high PM2.5 (in top 10%) and high Asthma (in top 10%) prevalence. The following map shows where they are.

2. Scatter plot (PM2.5 ~ Asthma)

From the scatter plot we can see that the apparent ‘fitness’ of the best-fit line is not good or satisfactory. This is because there are plenty of points far from the best-fit line between 8 µg/m3 to 9 µg/m3 of PM2.5. Maybe a curve can fit better rather than a straight line.

3. Linear Regression Analysis

## 
## Call:
## lm(formula = Asthma ~ PM2.5, data = PM25_Asthma)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -54.47 -25.89  -9.61  12.94 182.95 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -116.278     13.040  -8.917   <2e-16 ***
## PM2.5         19.862      1.534  12.950   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 37.49 on 1578 degrees of freedom
## Multiple R-squared:  0.09606,    Adjusted R-squared:  0.09549 
## F-statistic: 167.7 on 1 and 1578 DF,  p-value: < 2.2e-16

The result of regression analysis:
Asthma = 19.862 * PM2.5 - 116.278

An increase of 1 µg/m3 in PM2.5 is associated with an increase of ~20 (19.862) ED visits Asthma per 10,000 in Asthma; 9.6% of the variation in Asthma is explained by the variation in PM2.5. ( p-value: < 2e-16, statistically significant)

4. Residuals Analysis and Log Transformation

Residuals Analysis

Normality Analysis: From the residuals plot we can see that the mean of residuals is close to zero ( 0.0006923995), but there appears to be a skew to the density curve for the residuals, which might mean that we don’t have the conditions necessary to meaningfully interpret regression results on the data. This particular distribution looks skewed. Also, from the original scatter plot we can imagine that maybe a curve is more appropriate than a straight line to conduct the regression.

log(Asthma) ~ PM2.5

## 
## Call:
## lm(formula = log(Asthma) ~ PM2.5, data = PM25_Asthma)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.00402 -0.46479  0.03313  0.42298  1.75525 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.69234    0.22840   3.031  0.00248 ** 
## PM2.5        0.35633    0.02686  13.264  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6566 on 1578 degrees of freedom
## Multiple R-squared:  0.1003, Adjusted R-squared:  0.09974 
## F-statistic: 175.9 on 1 and 1578 DF,  p-value: < 2.2e-16

The result of log-regression analysis:
Log(Asthma) = 0.35633 * PM2.5 + 0.69234

An increase of 1 µg/m3 in PM2.5 is associated with an increase of 0.35633 log(ED visits Asthma per 10,000 in Asthma); 10% of the variation in log(Asthma) is explained by the variation in PM2.5. ( p-value: < 2e-16, statistically significant)

5. Log Model Residuals

Normality Analysis: From the residuals plot we can see that the mean of residuals is close to zero ( 1.864503e-16), and there is no skew to the density curve for the residuals. However, there is a small pit near 0. This shows the log transformation made the regression more reasonable. And also the regression curve is more like what we expect.

The map of Real_residuals
Real_residuals = Asthma - exp(model_log$fitted.values)

Here, I use the log-regression model (Log(Asthma) = 0.35633 * PM2.5 + 0.69234). So, I when I combine the residuals with the residuals with spatial infomation, I transfer the log(Asthma) to real Asthma by using exp() first. In the figure, residuals = measure Asthma - approximate variable(log model).

The census tract with the lowest “residual” is 6055200504 (County: Napa), the residual is -46.22854. From the identification of residual (real - approximation), we can know a low residual means in the context of Asthma estimation means that it is over-estimated. The reason that the census tract (6055200504) has lowest “residual” might be although the PM2.5 there is relatively high, there is a river (Napa River) which can fresh the air and mitigate Asthma to some extent. Also, there might be less hospitals for treating Asthma.